Random Fibonacci sequences and the number

For the familiar Fibonacci sequence (defined by , and for ), increases exponentially with at a rate given by the golden ratio . But for a simple modification with both additions and subtractions - the random Fibonacci sequences defined by , and for , , where each sign is independent and either or - with probability - it is not even obvious if should increase with . Our main result is that

with probability . Finding the number involves the theory of random matrix products, Stern-Brocot division of the real line, a fractal measure, a computer calculation, and a rounding error analysis to validate the computer calculation.

     

A new bound for the smallest

Let denote the number of primes and let denote the usual integral logarithm of . We prove that there are at least integer values of in the vicinity of with . This improves earlier bounds of Skewes, Lehman, and te Riele. We also plot more than 10000 values of in four different regions, including the regions discovered by Lehman, te Riele, and the authors of this paper, and a more distant region in the vicinity of , where appears to exceed by more than . The plots strongly suggest, although upper bounds derived to date for are not sufficient for a proof, that exceeds for at least integers in the vicinity of . If it is possible to improve our bound for by finding a sign change before , our first plot clearly delineates the potential candidates. Finally, we compute the logarithmic density of and find that as departs from the region in the vicinity of , the density is , and that it varies from this by no more than over the next integers. This should be compared to Rubinstein and Sarnak.

     

The Postage Stamp Problem: An algorithm to determine the

Given an integral ``stamp" basis with and a positive integer , we define the -range as

. For given and , the extremal basis has the largest possible extremal -range

We give an algorithm to determine the -range. We prove some properties of the -range formula, and we conjecture its form for the extremal -range. We consider parameter bases , where the basis elements are given functions of . For we conjecture the extremal parameter bases for .

     

Locking-free finite elements for the Reissner-Mindlin plate

Two new families of Reissner-Mindlin triangular finite elements are analyzed. One family, generalizing an element proposed by Zienkiewicz and Lefebvre, approximates (for the transverse displacement by continuous piecewise polynomials of degree , the rotation by continuous piecewise polynomials of degree plus bubble functions of degree , and projects the shear stress into the space of discontinuous piecewise polynomials of degree . The second family is similar to the first, but uses degree rather than degree continuous piecewise polynomials to approximate the rotation. We prove that for , the errors in the derivatives of the transverse displacement are bounded by and the errors in the rotation and its derivatives are bounded by and , respectively, for the first family, and by and , respectively, for the second family (with independent of the mesh size and plate thickness . These estimates are of optimal order for the second family, and so it is locking-free. For the first family, while the estimates for the derivatives of the transverse displacement are of optimal order, there is a deterioration of order in the approximation of the rotation and its derivatives for small, demonstrating locking of order . Numerical experiments using the lowest order elements of each family are presented to show their performance and the sharpness of the estimates. Additional experiments show the negative effects of eliminating the projection of the shear stress.

     

Class group frequencies of real quadratic function fields: The degree 4 case

The distribution of ideal class groups of is examined for degree-four monic polynomials when is a finite field of characteristic greater than 3 with or and is irreducible or has an irreducible cubic factor. Particular attention is paid to the distribution of the -Sylow part of the class group, and these results agree with those predicted using the Cohen-Lenstra heuristics to within about 1 part in 10000. An alternative set of conjectures specific to the cases under investigation is in even sharper agreement.

     

Extrapolation methods and derivatives of limits of sequences

Let be an infinite sequence whose limit or antilimit can be approximated very efficiently by applying a suitable extrapolation method E to . Assume that the and hence also are differentiable functions of some parameter , being the limit or antilimit of , and that we need to approximate . A direct way of achieving this would be by applying again a suitable extrapolation method E to the sequence , and this approach has often been used efficiently in various problems of practical importance. Unfortunately, as has been observed at least in some important cases, when and have essentially different asymptotic behaviors as , the approximations to produced by this approach, despite the fact that they are good, do not converge as quickly as those obtained for , and this is puzzling. In this paper we first give a rigorous mathematical explanation of this phenomenon for the cases in which E is the Richardson extrapolation process and E is a generalization of it, thus showing that the phenomenon has very little to do with numerics. Following that, we propose a procedure that amounts to first applying the extrapolation method E to and then differentiating the resulting approximations to , and we provide a thorough convergence and stability analysis in conjunction with the Richardson extrapolation process. It follows from this analysis that the new procedure for has practically the same convergence properties as E for . We show that a very efficient way of implementing the new procedure is by actually differentiating the recursion relations satisfied by the extrapolation method used, and we derive the necessary algorithm for the Richardson extrapolation process. We demonstrate the effectiveness of the new approach with numerical examples that also support the theory. We discuss the application of this approach to numerical integration in the presence of endpoint singularities. We also discuss briefly its application in conjunction with other extrapolation methods.

     

On the convergence of certain Gauss-type quadrature formulas for unbounded intervals

We consider the convergence of Gauss-type quadrature formulas for the integral , where is a weight function on the half line . The -point Gauss-type quadrature formulas are constructed such that they are exact in the set of Laurent polynomials }, where is a sequence of integers satisfying and . It is proved that under certain Carleman-type conditions for the weight and when or goes to , then convergence holds for all functions for which is integrable on . Some numerical experiments compare the convergence of these quadrature formulas with the convergence of the classical Gauss quadrature formulas for the half line.

     

Computation of relative class numbers of CM-fields by using Hecke -functions

We develop an efficient technique for computing values at of Hecke -functions. We apply this technique to the computation of relative class numbers of non-abelian CM-fields which are abelian extensions of some totally real subfield . We note that the smaller the degree of the more efficient our technique is. In particular, our technique is very efficient whenever instead of simply choosing (the maximal totally real subfield of ) we can choose real quadratic. We finally give examples of computations of relative class numbers of several dihedral CM-fields of large degrees and of several quaternion octic CM-fields with large discriminants.

     

On fundamental domains of arithmetic Fuchsian groups

Let be a totally real algebraic number field and an order in a quaternion algebra over . Assume that the group of units in with reduced norm equal to is embedded into as an arithmetic Fuchsian group. It is shown how Ford's algorithm can be effectively applied in order to determine a fundamental domain of as well as a complete system of generators of .

     

The number of primes is finite

For a positive integer let and let . The number of primes of the form is finite, because if , then is divisible by . The heuristic argument is given by which there exists a prime such that for all large ; a computer check however shows that this prime has to be greater than . The conjecture that the numbers are squarefree is not true because .

     

Families of irreducible polynomials of Gaussian periods and matrices of cyclotomic numbers

Given an odd prime we show a way to construct large families of polynomials , , where is a set of primes of the form mod and is the irreducible polynomial of the Gaussian periods of degree in . Examples of these families when are worked in detail. We also show, given an integer and a prime mod , how to represent by matrices the Gaussian periods of degree in , and how to calculate in a simple way, with the help of a computer, irreducible polynomials for elements of .

     

Nonexistence conditions of a solution for the congruence

We obtain nonexistence conditions of a solution for of the congruence , where , and are integers, and is a prime power. We give nonexistence conditions of the form for , , , , , and of the form for , , , . Furthermore, we complete some tables concerned with Waring's problem in -adic fields that were computed by Hardy and Littlewood.

     

Using number fields to compute logarithms in finite fields

We describe an adaptation of the number field sieve to the problem of computing logarithms in a finite field. We conjecture that the running time of the algorithm, when restricted to finite fields of an arbitrary but fixed degree, is where is the cardinality of the field, and the is for . The number field sieve factoring algorithm is conjectured to factor a number the size of in the same amount of time.

     

Salem numbers of negative trace

We prove that, for all , there are Salem numbers of degree and trace , and that the number of such Salem numbers is . As a consequence, it follows that the number of totally positive algebraic integers of degree and trace is also .

     

Error estimates in in covolume methods for elliptic and parabolic problems: A unified approach

In this paper we consider covolume or finite volume element methods for variable coefficient elliptic and parabolic problems on convex smooth domains in the plane. We introduce a general approach for connecting these methods with finite element method analysis. This unified approach is used to prove known convergence results in the norms and new results in the max-norm. For the elliptic problems we demonstrate that the error between the exact solution and the approximate solution in the maximum norm is in the linear element case. Furthermore, the maximum norm error in the gradient is shown to be of first order. Similar results hold for the parabolic problems.

     

Uniform convergence results for the mortar finite element method

The mortar finite element is an example of a non-conforming method which can be used to decompose and re-compose a domain into subdomains without requiring compatibility between the meshes on the separate components. We obtain stability and convergence results for this method that are uniform in terms of both the degree and the mesh used, without assuming quasiuniformity for the meshes. Our results establish that the method is optimal when non-quasiuniform or methods are used. Such methods are essential in practice for good rates of convergence when the interface passes through a corner of the domain. We also give an error estimate for when the version is used. Numerical results for and mortar FEMs show that these methods behave as well as conforming FEMs. An extension theorem is also proved.

     

Stability of Runge-Kutta methods for quasilinear parabolic problems

We consider a quasilinear parabolic problem

where , , is a family of sectorial operators in a Banach space with fixed domain . This problem is discretized in time by means of a strongly A()-stable, , Runge-Kutta method. We prove that the resulting discretization is stable, under some natural assumptions on the dependence of with respect to . Our results are useful for studying in norms, , many problems arising in applications. Some auxiliary results for time-dependent parabolic problems are also provided.

     

Tables of curves with many points

These tables record results on curves with many points over finite fields. For relatively small genus () and a small power of or we give in two tables the best presently known bounds for , the maximum number of rational points on a smooth absolutely irreducible projective curve of genus over a field of cardinality . In additional tables we list for a given pair the type of construction of the best curve so far, and we give a reference to the literature where such a curve can be found.

     

Computational scales of Sobolev norms with application to preconditioning

This paper provides a framework for developing computationally efficient multilevel preconditioners and representations for Sobolev norms. Specifically, given a Hilbert space and a nested sequence of subspaces , we construct operators which are spectrally equivalent to those of the form . Here , , are positive numbers and is the orthogonal projector onto with . We first present abstract results which show when is spectrally equivalent to a similarly constructed operator defined in terms of an approximation of , for . We show that these results lead to efficient preconditioners for discretizations of differential and pseudo-differential operators of positive and negative order. These results extend to sums of operators. For example, singularly perturbed problems such as can be preconditioned uniformly independently of the parameter . We also show how to precondition an operator which results from Tikhonov regularization of a problem with noisy data. Finally, we describe how the technique provides computationally efficient bounded discrete extensions which have applications to domain decomposition.